How do you use the definition of the derivative to find the derivative of the function #f(x) = 6 x + 2sqrt{x}#?

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Answer 1 · 2018-04-09T11:12:16

Please look below.

Defintion of a derivative:

#f'(x) = lim_(hrarr0)(f(x+h)-f(x))/h#

#f(x) = 6x + 2sqrtx#
#f(x+h) = 6(x+h) + 2sqrt(x+h)#

#f'(x) = lim_(hrarr0)(6(x+h) + 2sqrt(x+h)-(6x + 2sqrtx))/h#

#f'(x) = lim_(hrarr0)(6h + 2(sqrt(x+h) - sqrt(x)))/h#

#f'(x) = 6+ 2lim_(hrarr0)(sqrt(x+h) - sqrt(x))/h#

#f'(x) = 6+ 2lim_(hrarr0)(sqrt(x+h) - sqrt(x))/h xx (sqrt(x+h) + sqrt(x))/(sqrt(x+h) + sqrt(x))#

#f'(x) = 6+2lim_(hrarr0)(x+h - x)/(h(sqrt(x+h) + sqrt(x)))#

#f'(x) = 6+2lim_(hrarr0)(h )/(h(sqrt(x+h) + sqrt(x)))#

#f'(x) = 6+2lim_(hrarr0)(1 )/(sqrt(x+h) + sqrt(x))#

#f'(x) = 6 + 2 xx 1/(sqrtx+sqrtx)#

#f'(x) = 6 + 1/sqrtx#